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Magma
magma: G := TransitiveGroup(35, 50);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $A_7:F_5$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,35,22,8,16,5,32,23,6,20,2,33,21,10,17,3,31,25,7,18)(4,34,24,9,19)(11,30,12,28)(13,26,15,27)(14,29), (1,8,20,2,9,16,3,10,17,4,6,18,5,7,19)(11,23,15,22,14,21,13,25,12,24)(26,33,30,32,29,31,28,35,27,34) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ $5040$: $S_7$ $10080$: 28T360 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $S_7$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $5^{7}$ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $2^{14},1^{7}$ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ | |
| $4^{7},2,1^{5}$ | $105$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(21,31)(22,34,25,33) (23,32,24,35)(27,29,30,28)$ | |
| $4^{7},2,1^{5}$ | $105$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(21,31)(22,33,25,34) (23,35,24,32)(27,28,30,29)$ | |
| $10^{2},5,2^{4},1^{2}$ | $2520$ | $10$ | $( 1,15, 6,20,26, 5,11,10,16,30)( 2,14, 7,19,27, 4,12, 9,17,29)( 3,13, 8,18,28) (21,25)(22,24)(31,35)(32,34)$ | |
| $5^{7}$ | $2016$ | $5$ | $( 1,12, 8,19,30)( 2,13, 9,20,26)( 3,14,10,16,27)( 4,15, 6,17,28) ( 5,11, 7,18,29)(21,22,23,24,25)(31,32,33,34,35)$ | |
| $5^{5},1^{10}$ | $504$ | $5$ | $( 1,11, 6,16,26)( 2,12, 7,17,27)( 3,13, 8,18,28)( 4,14, 9,19,29) ( 5,15,10,20,30)$ | |
| $20,5,4^{2},2$ | $2520$ | $20$ | $( 1,17,15,29, 6, 2,20,14,26, 7, 5,19,11,27,10, 4,16,12,30, 9)( 3,18,13,28, 8) (21,32,25,34)(22,35,24,31)(23,33)$ | |
| $20,5,4^{2},2$ | $2520$ | $20$ | $( 1,20,13,29, 6, 5,18,14,26,10, 3,19,11,30, 8, 4,16,15,28, 9)( 2,17,12,27, 7) (21,35,23,34)(22,32)(24,31,25,33)$ | |
| $10^{2},5^{3}$ | $420$ | $10$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,32,18,34,20,31,17,33,19,35 )(21,27,23,29,25,26,22,28,24,30)$ | |
| $2^{10},1^{15}$ | $105$ | $2$ | $(16,31)(17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$ | |
| $2^{16},1^{3}$ | $525$ | $2$ | $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,15)(12,14)(16,35)(17,34)(18,33)(19,32)(20,31) (21,30)(22,29)(23,28)(24,27)(25,26)$ | |
| $4^{5},2^{7},1$ | $3150$ | $4$ | $( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,27,31,22)(17,26,32,21) (18,30,33,25)(19,29,34,24)(20,28,35,23)$ | |
| $20,10,5$ | $2520$ | $20$ | $( 1, 5, 4, 3, 2)( 6,15, 9,13, 7,11,10,14, 8,12)(16,30,34,23,17,26,35,24,18,27, 31,25,19,28,32,21,20,29,33,22)$ | |
| $4^{5},2^{5},1^{5}$ | $630$ | $4$ | $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,26,31,21)(17,27,32,22)(18,28,33,23) (19,29,34,24)(20,30,35,25)$ | |
| $4^{8},1^{3}$ | $1050$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(16,21,31,26)(17,24,35,28)(18,22,34,30) (19,25,33,27)(20,23,32,29)$ | |
| $4^{8},1^{3}$ | $1050$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(16,21,31,26)(17,23,35,29)(18,25,34,27) (19,22,33,30)(20,24,32,28)$ | |
| $15,5^{4}$ | $280$ | $15$ | $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,18,20,17,19)(21,23,25,22,24) (26,28,30,27,29)(31,33,35,32,34)$ | |
| $3^{5},1^{20}$ | $70$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ | |
| $6^{2},3,2^{8},1^{4}$ | $350$ | $6$ | $( 1, 9,11, 4, 6,14)( 2, 8,12, 3, 7,13)( 5,10,15)(16,19)(17,18)(21,24)(22,23) (26,29)(27,28)(31,34)(32,33)$ | |
| $12,4^{4},3,2,1^{2}$ | $2100$ | $12$ | $( 1,13, 7, 5,11, 8, 2,15, 6, 3,12,10)( 4,14, 9)(16,23,17,25)(18,22,20,21) (19,24)(26,28,27,30)(31,33,32,35)$ | |
| $12,4^{4},3,2,1^{2}$ | $2100$ | $12$ | $( 1,14, 8, 5,11, 9, 3,15, 6, 4,13,10)( 2,12, 7)(16,24,18,25)(17,22) (19,23,20,21)(26,29,28,30)(31,34,33,35)$ | |
| $4^{7},2^{3},1$ | $525$ | $4$ | $( 1, 4, 5, 2)( 6,14,10,12)( 7,11, 9,15)( 8,13)(16,34,20,32)(17,31,19,35) (18,33)(21,29,25,27)(22,26,24,30)(23,28)$ | |
| $4^{7},2^{3},1$ | $525$ | $4$ | $( 1, 3, 4, 2)( 6,13, 9,12)( 7,11, 8,14)(10,15)(16,33,19,32)(17,31,18,34) (20,35)(21,28,24,27)(22,26,23,29)(25,30)$ | |
| $15^{2},5$ | $1120$ | $15$ | $( 1, 4, 2, 5, 3)( 6,19,27,10,18,26, 9,17,30, 8,16,29, 7,20,28)(11,34,22,15,33, 21,14,32,25,13,31,24,12,35,23)$ | |
| $3^{10},1^{5}$ | $280$ | $3$ | $( 6,16,26)( 7,17,27)( 8,18,28)( 9,19,29)(10,20,30)(11,31,21)(12,32,22) (13,33,23)(14,34,24)(15,35,25)$ | |
| $6^{4},3^{2},2^{2},1$ | $1400$ | $6$ | $( 1, 3)( 4, 5)( 6,18,26, 8,16,28)( 7,17,27)( 9,20,29,10,19,30)(11,33,21,13,31, 23)(12,32,22)(14,35,24,15,34,25)$ | |
| $12^{2},6,4,1$ | $4200$ | $12$ | $( 1, 2, 4, 3)( 6,22,19,13,26,32, 9,23,16,12,29,33)( 7,24,18,11,27,34, 8,21,17, 14,28,31)(10,25,20,15,30,35)$ | |
| $12^{2},6,4,1$ | $4200$ | $12$ | $( 1, 5, 2, 3)( 6,25,17,13,26,35, 7,23,16,15,27,33)( 8,21,20,12,28,31,10,22,18, 11,30,32)( 9,24,19,14,29,34)$ | |
| $6^{2},3,2^{10}$ | $1050$ | $6$ | $( 1,13, 6, 3,11, 8)( 2,12, 7)( 4,15, 9, 5,14,10)(16,33)(17,32)(18,31)(19,35) (20,34)(21,28)(22,27)(23,26)(24,30)(25,29)$ | |
| $15,10^{2}$ | $840$ | $30$ | $( 1,14, 7, 5,13, 6, 4,12,10, 3,11, 9, 2,15, 8)(16,34,17,35,18,31,19,32,20,33) (21,29,22,30,23,26,24,27,25,28)$ | |
| $3^{5},2^{10}$ | $210$ | $6$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)(16,31)(17,32)(18,33)(19,34) (20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$ | |
| $12,4^{5},3$ | $2100$ | $12$ | $( 1, 9,13, 5, 6,14, 3,10,11, 4, 8,15)( 2, 7,12)(16,24,33,30)(17,22,32,27) (18,25,31,29)(19,23,35,26)(20,21,34,28)$ | |
| $12,4^{5},3$ | $2100$ | $12$ | $( 1, 8,12, 5, 6,13, 2,10,11, 3, 7,15)( 4, 9,14)(16,23,32,30)(17,25,31,28) (18,22,35,26)(19,24,34,29)(20,21,33,27)$ | |
| $7^{5}$ | $720$ | $7$ | $( 1,21,31, 6,11,16,26)( 2,22,32, 7,12,17,27)( 3,23,33, 8,13,18,28) ( 4,24,34, 9,14,19,29)( 5,25,35,10,15,20,30)$ | |
| $35$ | $1440$ | $35$ | $( 1,22,33, 9,15,16,27, 3,24,35, 6,12,18,29, 5,21,32, 8,14,20,26, 2,23,34,10, 11,17,28, 4,25,31, 7,13,19,30)$ | |
| $35$ | $1440$ | $35$ | $( 1,23,35, 7,14,16,28, 5,22,34, 6,13,20,27, 4,21,33,10,12,19,26, 3,25,32, 9, 11,18,30, 2,24,31, 8,15,17,29)$ | |
| $14^{2},7$ | $3600$ | $14$ | $( 1,21,31, 6,11,16,26)( 2,25,32,10,12,20,27, 5,22,35, 7,15,17,30) ( 3,24,33, 9,13,19,28, 4,23,34, 8,14,18,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 50400.h | magma: IdentifyGroup(G);
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| Character table: | 39 x 39 character table |
magma: CharacterTable(G);